/// @file
/// Special Euclidean group SE(3) - rotation and translation in 3d.

#ifndef SOPHUS_SE3_HPP
#define SOPHUS_SE3_HPP

#include "so3.hpp"

namespace Sophus
{
  template <class Scalar_, int Options = 0>
  class SE3;
  using SE3d = SE3<double>;
  using SE3f = SE3<float>;
} // namespace Sophus

namespace Eigen
{
  namespace internal
  {
    template <class Scalar_, int Options>
    struct traits<Sophus::SE3<Scalar_, Options>>
    {
      using Scalar = Scalar_;
      using TranslationType = Sophus::Vector3<Scalar, Options>;
      using SO3Type = Sophus::SO3<Scalar, Options>;
    };

    template <class Scalar_, int Options>
    struct traits<Map<Sophus::SE3<Scalar_>, Options>>
        : traits<Sophus::SE3<Scalar_, Options>>
    {
      using Scalar = Scalar_;
      using TranslationType = Map<Sophus::Vector3<Scalar>, Options>;
      using SO3Type = Map<Sophus::SO3<Scalar>, Options>;
    };

    template <class Scalar_, int Options>
    struct traits<Map<Sophus::SE3<Scalar_> const, Options>>
        : traits<Sophus::SE3<Scalar_, Options> const>
    {
      using Scalar = Scalar_;
      using TranslationType = Map<Sophus::Vector3<Scalar> const, Options>;
      using SO3Type = Map<Sophus::SO3<Scalar> const, Options>;
    };
  } // namespace internal
} // namespace Eigen

namespace Sophus
{
  /// SE3 base type - implements SE3 class but is storage agnostic.
  ///
  /// SE(3) is the group of rotations  and translation in 3d. It is the
  /// semi-direct product of SO(3) and the 3d Euclidean vector space.  The class
  /// is represented using a composition of SO3  for rotation and a one 3-vector
  /// for translation.
  ///
  /// SE(3) is neither compact, nor a commutative group.
  ///
  /// See SO3 for more details of the rotation representation in 3d.
  ///
  template <class Derived>
  class SE3Base
  {
  public:
    using Scalar = typename Eigen::internal::traits<Derived>::Scalar;
    using TranslationType =
        typename Eigen::internal::traits<Derived>::TranslationType;
    using SO3Type = typename Eigen::internal::traits<Derived>::SO3Type;
    using QuaternionType = typename SO3Type::QuaternionType;
    /// Degrees of freedom of manifold, number of dimensions in tangent space
    /// (three for translation, three for rotation).
    static int constexpr DoF = 6;
    /// Number of internal parameters used (4-tuple for quaternion, three for
    /// translation).
    static int constexpr num_parameters = 7;
    /// Group transformations are 4x4 matrices.
    static int constexpr N = 4;
    using Transformation = Matrix<Scalar, N, N>;
    using Point = Vector3<Scalar>;
    using HomogeneousPoint = Vector4<Scalar>;
    using Line = ParametrizedLine3<Scalar>;
    using Tangent = Vector<Scalar, DoF>;
    using Adjoint = Matrix<Scalar, DoF, DoF>;

    /// For binary operations the return type is determined with the
    /// ScalarBinaryOpTraits feature of Eigen. This allows mixing concrete and Map
    /// types, as well as other compatible scalar types such as Ceres::Jet and
    /// double scalars with SE3 operations.
    template <typename OtherDerived>
    using ReturnScalar = typename Eigen::ScalarBinaryOpTraits<
        Scalar, typename OtherDerived::Scalar>::ReturnType;

    template <typename OtherDerived>
    using SE3Product = SE3<ReturnScalar<OtherDerived>>;

    template <typename PointDerived>
    using PointProduct = Vector3<ReturnScalar<PointDerived>>;

    template <typename HPointDerived>
    using HomogeneousPointProduct = Vector4<ReturnScalar<HPointDerived>>;

    /// Adjoint transformation
    ///
    /// This function return the adjoint transformation ``Ad`` of the group
    /// element ``A`` such that for all ``x`` it holds that
    /// ``hat(Ad_A * x) = A * hat(x) A^{-1}``. See hat-operator below.
    ///
    SOPHUS_FUNC Adjoint Adj() const
    {
      Sophus::Matrix3<Scalar> const R = so3().matrix();
      Adjoint res;
      res.block(0, 0, 3, 3) = R;
      res.block(3, 3, 3, 3) = R;
      res.block(0, 3, 3, 3) = SO3<Scalar>::hat(translation()) * R;
      res.block(3, 0, 3, 3) = Matrix3<Scalar>::Zero(3, 3);

      return res;
    }

    /// Extract rotation angle about canonical X-axis
    ///
    Scalar angleX() const { return so3().angleX(); }

    /// Extract rotation angle about canonical Y-axis
    ///
    Scalar angleY() const { return so3().angleY(); }

    /// Extract rotation angle about canonical Z-axis
    ///
    Scalar angleZ() const { return so3().angleZ(); }

    /// Returns copy of instance casted to NewScalarType.
    ///
    template <class NewScalarType>
    SOPHUS_FUNC SE3<NewScalarType> cast() const
    {
      return SE3<NewScalarType>(so3().template cast<NewScalarType>(),
                                translation().template cast<NewScalarType>());
    }

    /// Returns derivative of  this * exp(x)  wrt x at x=0.
    ///
    SOPHUS_FUNC Matrix<Scalar, num_parameters, DoF> Dx_this_mul_exp_x_at_0()
        const
    {
      Matrix<Scalar, num_parameters, DoF> J;
      Eigen::Quaternion<Scalar> const q = unit_quaternion();
      Scalar const c0 = Scalar(0.5) * q.w();
      Scalar const c1 = Scalar(0.5) * q.z();
      Scalar const c2 = -c1;
      Scalar const c3 = Scalar(0.5) * q.y();
      Scalar const c4 = Scalar(0.5) * q.x();
      Scalar const c5 = -c4;
      Scalar const c6 = -c3;
      Scalar const c7 = q.w() * q.w();
      Scalar const c8 = q.x() * q.x();
      Scalar const c9 = q.y() * q.y();
      Scalar const c10 = -c9;
      Scalar const c11 = q.z() * q.z();
      Scalar const c12 = -c11;
      Scalar const c13 = Scalar(2) * q.w();
      Scalar const c14 = c13 * q.z();
      Scalar const c15 = Scalar(2) * q.x();
      Scalar const c16 = c15 * q.y();
      Scalar const c17 = c13 * q.y();
      Scalar const c18 = c15 * q.z();
      Scalar const c19 = c7 - c8;
      Scalar const c20 = c13 * q.x();
      Scalar const c21 = Scalar(2) * q.y() * q.z();

      J(0, 0) = 0;
      J(0, 1) = 0;
      J(0, 2) = 0;
      J(0, 3) = c0;
      J(0, 4) = c2;
      J(0, 5) = c3;
      J(1, 0) = 0;
      J(1, 1) = 0;
      J(1, 2) = 0;
      J(1, 3) = c1;
      J(1, 4) = c0;
      J(1, 5) = c5;
      J(2, 0) = 0;
      J(2, 1) = 0;
      J(2, 2) = 0;
      J(2, 3) = c6;
      J(2, 4) = c4;
      J(2, 5) = c0;
      J(3, 0) = 0;
      J(3, 1) = 0;
      J(3, 2) = 0;
      J(3, 3) = c5;
      J(3, 4) = c6;
      J(3, 5) = c2;
      J(4, 0) = c10 + c12 + c7 + c8;
      J(4, 1) = -c14 + c16;
      J(4, 2) = c17 + c18;
      J(4, 3) = 0;
      J(4, 4) = 0;
      J(4, 5) = 0;
      J(5, 0) = c14 + c16;
      J(5, 1) = c12 + c19 + c9;
      J(5, 2) = -c20 + c21;
      J(5, 3) = 0;
      J(5, 4) = 0;
      J(5, 5) = 0;
      J(6, 0) = -c17 + c18;
      J(6, 1) = c20 + c21;
      J(6, 2) = c10 + c11 + c19;
      J(6, 3) = 0;
      J(6, 4) = 0;
      J(6, 5) = 0;

      return J;
    }

    /// Returns group inverse.
    ///
    SOPHUS_FUNC SE3<Scalar> inverse() const
    {
      SO3<Scalar> invR = so3().inverse();
      return SE3<Scalar>(invR, invR * (translation() * Scalar(-1)));
    }

    /// Logarithmic map
    ///
    /// Computes the logarithm, the inverse of the group exponential which maps
    /// element of the group (rigid body transformations) to elements of the
    /// tangent space (twist).
    ///
    /// To be specific, this function computes ``vee(logmat(.))`` with
    /// ``logmat(.)`` being the matrix logarithm and ``vee(.)`` the vee-operator
    /// of SE(3).
    ///
    SOPHUS_FUNC Tangent log() const
    {
      // For the derivation of the logarithm of SE(3), see
      // J. Gallier, D. Xu, "Computing exponentials of skew symmetric matrices
      // and logarithms of orthogonal matrices", IJRA 2002.
      // https:///pdfs.semanticscholar.org/cfe3/e4b39de63c8cabd89bf3feff7f5449fc981d.pdf
      // (Sec. 6., pp. 8)
      using std::abs;
      using std::cos;
      using std::sin;
      Tangent upsilon_omega;
      auto omega_and_theta = so3().logAndTheta();
      Scalar theta = omega_and_theta.theta;
      upsilon_omega.template tail<3>() = omega_and_theta.tangent;
      Matrix3<Scalar> const Omega =
          SO3<Scalar>::hat(upsilon_omega.template tail<3>());

      if (abs(theta) < Constants<Scalar>::epsilon())
      {
        Matrix3<Scalar> const V_inv = Matrix3<Scalar>::Identity() -
                                      Scalar(0.5) * Omega +
                                      Scalar(1. / 12.) * (Omega * Omega);

        upsilon_omega.template head<3>() = V_inv * translation();
      }
      else
      {
        Scalar const half_theta = Scalar(0.5) * theta;

        Matrix3<Scalar> const V_inv =
            (Matrix3<Scalar>::Identity() - Scalar(0.5) * Omega +
             (Scalar(1) -
              theta * cos(half_theta) / (Scalar(2) * sin(half_theta))) /
                 (theta * theta) * (Omega * Omega));
        upsilon_omega.template head<3>() = V_inv * translation();
      }

      return upsilon_omega;
    }

    /// It re-normalizes the SO3 element.
    ///
    /// Note: Because of the class invariant of SO3, there is typically no need to
    /// call this function directly.
    ///
    SOPHUS_FUNC void normalize() { so3().normalize(); }

    /// Returns 4x4 matrix representation of the instance.
    ///
    /// It has the following form:
    ///
    ///   | R t |
    ///   | o 1 |
    ///
    /// where ``R`` is a 3x3 rotation matrix, ``t`` a translation 3-vector and
    /// ``o`` a 3-column vector of zeros.
    ///
    SOPHUS_FUNC Transformation matrix() const
    {
      Transformation homogenious_matrix;
      homogenious_matrix.template topLeftCorner<3, 4>() = matrix3x4();
      homogenious_matrix.row(3) =
          Matrix<Scalar, 1, 4>(Scalar(0), Scalar(0), Scalar(0), Scalar(1));

      return homogenious_matrix;
    }

    /// Returns the significant first three rows of the matrix above.
    ///
    SOPHUS_FUNC Matrix<Scalar, 3, 4> matrix3x4() const
    {
      Matrix<Scalar, 3, 4> matrix;
      matrix.template topLeftCorner<3, 3>() = rotationMatrix();
      matrix.col(3) = translation();

      return matrix;
    }

    /// Assignment operator.
    ///
    SOPHUS_FUNC SE3Base &operator=(SE3Base const &other) = default;

    /// Assignment-like operator from OtherDerived.
    ///
    template <class OtherDerived>
    SOPHUS_FUNC SE3Base<Derived> &operator=(SE3Base<OtherDerived> const &other)
    {
      so3() = other.so3();
      translation() = other.translation();

      return *this;
    }

    /// Group multiplication, which is rotation concatenation.
    ///
    template <typename OtherDerived>
    SOPHUS_FUNC SE3Product<OtherDerived> operator*(
        SE3Base<OtherDerived> const &other) const
    {
      return SE3Product<OtherDerived>(
          so3() * other.so3(), translation() + so3() * other.translation());
    }

    /// Group action on 3-points.
    ///
    /// This function rotates and translates a three dimensional point ``p`` by
    /// the SE(3) element ``bar_T_foo = (bar_R_foo, t_bar)`` (= rigid body
    /// transformation):
    ///
    ///   ``p_bar = bar_R_foo * p_foo + t_bar``.
    ///
    template <typename PointDerived,
              typename = typename std::enable_if<
                  IsFixedSizeVector<PointDerived, 3>::value>::type>
    SOPHUS_FUNC PointProduct<PointDerived> operator*(
        Eigen::MatrixBase<PointDerived> const &p) const
    {
      return so3() * p + translation();
    }

    /// Group action on homogeneous 3-points. See above for more details.
    ///
    template <typename HPointDerived,
              typename = typename std::enable_if<
                  IsFixedSizeVector<HPointDerived, 4>::value>::type>
    SOPHUS_FUNC HomogeneousPointProduct<HPointDerived> operator*(
        Eigen::MatrixBase<HPointDerived> const &p) const
    {
      const PointProduct<HPointDerived> tp =
          so3() * p.template head<3>() + p(3) * translation();
      return HomogeneousPointProduct<HPointDerived>(tp(0), tp(1), tp(2), p(3));
    }

    /// Group action on lines.
    ///
    /// This function rotates and translates a parametrized line
    /// ``l(t) = o + t * d`` by the SE(3) element:
    ///
    /// Origin is transformed using SE(3) action
    /// Direction is transformed using rotation part
    ///
    SOPHUS_FUNC Line operator*(Line const &l) const
    {
      return Line((*this) * l.origin(), so3() * l.direction());
    }

    /// In-place group multiplication. This method is only valid if the return
    /// type of the multiplication is compatible with this SE3's Scalar type.
    ///
    template <typename OtherDerived,
              typename = typename std::enable_if<
                  std::is_same<Scalar, ReturnScalar<OtherDerived>>::value>::type>
    SOPHUS_FUNC SE3Base<Derived> &operator*=(SE3Base<OtherDerived> const &other)
    {
      *static_cast<Derived *>(this) = *this * other;
      return *this;
    }

    /// Returns rotation matrix.
    ///
    SOPHUS_FUNC Matrix3<Scalar> rotationMatrix() const { return so3().matrix(); }

    /// Mutator of SO3 group.
    ///
    SOPHUS_FUNC SO3Type &so3() { return static_cast<Derived *>(this)->so3(); }

    /// Accessor of SO3 group.
    ///
    SOPHUS_FUNC SO3Type const &so3() const
    {
      return static_cast<const Derived *>(this)->so3();
    }

    /// Takes in quaternion, and normalizes it.
    ///
    /// Precondition: The quaternion must not be close to zero.
    ///
    SOPHUS_FUNC void setQuaternion(Eigen::Quaternion<Scalar> const &quat)
    {
      so3().setQuaternion(quat);
    }

    /// Sets ``so3`` using ``rotation_matrix``.
    ///
    /// Precondition: ``R`` must be orthogonal and ``det(R)=1``.
    ///
    SOPHUS_FUNC void setRotationMatrix(Matrix3<Scalar> const &R)
    {
      SOPHUS_ENSURE(isOrthogonal(R), "R is not orthogonal:\n %", R);
      SOPHUS_ENSURE(R.determinant() > Scalar(0), "det(R) is not positive: %",
                    R.determinant());
      so3().setQuaternion(Eigen::Quaternion<Scalar>(R));
    }

    /// Returns internal parameters of SE(3).
    ///
    /// It returns (q.imag[0], q.imag[1], q.imag[2], q.real, t[0], t[1], t[2]),
    /// with q being the unit quaternion, t the translation 3-vector.
    ///
    SOPHUS_FUNC Sophus::Vector<Scalar, num_parameters> params() const
    {
      Sophus::Vector<Scalar, num_parameters> p;
      p << so3().params(), translation();

      return p;
    }

    /// Mutator of translation vector.
    ///
    SOPHUS_FUNC TranslationType &translation()
    {
      return static_cast<Derived *>(this)->translation();
    }

    /// Accessor of translation vector
    ///
    SOPHUS_FUNC TranslationType const &translation() const
    {
      return static_cast<Derived const *>(this)->translation();
    }

    /// Accessor of unit quaternion.
    ///
    SOPHUS_FUNC QuaternionType const &unit_quaternion() const
    {
      return this->so3().unit_quaternion();
    }
  };

  /// SE3 using default storage; derived from SE3Base.
  template <class Scalar_, int Options>
  class SE3 : public SE3Base<SE3<Scalar_, Options>>
  {
    using Base = SE3Base<SE3<Scalar_, Options>>;

  public:
    static int constexpr DoF = Base::DoF;
    static int constexpr num_parameters = Base::num_parameters;

    using Scalar = Scalar_;
    using Transformation = typename Base::Transformation;
    using Point = typename Base::Point;
    using HomogeneousPoint = typename Base::HomogeneousPoint;
    using Tangent = typename Base::Tangent;
    using Adjoint = typename Base::Adjoint;
    using SO3Member = SO3<Scalar, Options>;
    using TranslationMember = Vector3<Scalar, Options>;

    EIGEN_MAKE_ALIGNED_OPERATOR_NEW

    /// Default constructor initializes rigid body motion to the identity.
    ///
    SOPHUS_FUNC SE3();

    /// Copy constructor
    ///
    SOPHUS_FUNC SE3(SE3 const &other) = default;

    /// Copy-like constructor from OtherDerived.
    ///
    template <class OtherDerived>
    SOPHUS_FUNC SE3(SE3Base<OtherDerived> const &other)
        : so3_(other.so3()), translation_(other.translation())
    {
      static_assert(std::is_same<typename OtherDerived::Scalar, Scalar>::value,
                    "must be same Scalar type");
    }

    /// Constructor from SO3 and translation vector
    ///
    template <class OtherDerived, class D>
    SOPHUS_FUNC SE3(SO3Base<OtherDerived> const &so3,
                    Eigen::MatrixBase<D> const &translation)
        : so3_(so3), translation_(translation)
    {
      static_assert(std::is_same<typename OtherDerived::Scalar, Scalar>::value,
                    "must be same Scalar type");
      static_assert(std::is_same<typename D::Scalar, Scalar>::value,
                    "must be same Scalar type");
    }

    /// Constructor from rotation matrix and translation vector
    ///
    /// Precondition: Rotation matrix needs to be orthogonal with determinant
    ///               of 1.
    ///
    SOPHUS_FUNC
    SE3(Matrix3<Scalar> const &rotation_matrix, Point const &translation)
        : so3_(rotation_matrix), translation_(translation) {}

    /// Constructor from quaternion and translation vector.
    ///
    /// Precondition: ``quaternion`` must not be close to zero.
    ///
    SOPHUS_FUNC SE3(Eigen::Quaternion<Scalar> const &quaternion,
                    Point const &translation)
        : so3_(quaternion), translation_(translation) {}

    /// Constructor from 4x4 matrix
    ///
    /// Precondition: Rotation matrix needs to be orthogonal with determinant
    ///               of 1. The last row must be ``(0, 0, 0, 1)``.
    ///
    SOPHUS_FUNC explicit SE3(Matrix4<Scalar> const &T)
        : so3_(T.template topLeftCorner<3, 3>()),
          translation_(T.template block<3, 1>(0, 3))
    {
      SOPHUS_ENSURE((T.row(3) - Matrix<Scalar, 1, 4>(Scalar(0), Scalar(0),
                                                     Scalar(0), Scalar(1)))
                            .squaredNorm() < Constants<Scalar>::epsilon(),
                    "Last row is not (0,0,0,1), but (%).", T.row(3));
    }

    /// This provides unsafe read/write access to internal data. SO(3) is
    /// represented by an Eigen::Quaternion (four parameters). When using direct
    /// write access, the user needs to take care of that the quaternion stays
    /// normalized.
    ///
    SOPHUS_FUNC Scalar *data()
    {
      // so3_ and translation_ are laid out sequentially with no padding
      return so3_.data();
    }

    /// Const version of data() above.
    ///
    SOPHUS_FUNC Scalar const *data() const
    {
      // so3_ and translation_ are laid out sequentially with no padding
      return so3_.data();
    }

    /// Mutator of SO3
    ///
    SOPHUS_FUNC SO3Member &so3() { return so3_; }

    /// Accessor of SO3
    ///
    SOPHUS_FUNC SO3Member const &so3() const { return so3_; }

    /// Mutator of translation vector
    ///
    SOPHUS_FUNC TranslationMember &translation() { return translation_; }

    /// Accessor of translation vector
    ///
    SOPHUS_FUNC TranslationMember const &translation() const
    {
      return translation_;
    }

    /// Returns derivative of exp(x) wrt. x.
    ///
    SOPHUS_FUNC static Sophus::Matrix<Scalar, num_parameters, DoF> Dx_exp_x(
        Tangent const &upsilon_omega)
    {
      using std::cos;
      using std::pow;
      using std::sin;
      using std::sqrt;
      Sophus::Matrix<Scalar, num_parameters, DoF> J;
      Sophus::Vector<Scalar, 3> upsilon = upsilon_omega.template head<3>();
      Sophus::Vector<Scalar, 3> omega = upsilon_omega.template tail<3>();

      Scalar const c0 = omega[0] * omega[0];
      Scalar const c1 = omega[1] * omega[1];
      Scalar const c2 = omega[2] * omega[2];
      Scalar const c3 = c0 + c1 + c2;
      Scalar const o(0);
      Scalar const h(0.5);
      Scalar const i(1);

      if (c3 < Constants<Scalar>::epsilon())
      {
        Scalar const ux = Scalar(0.5) * upsilon[0];
        Scalar const uy = Scalar(0.5) * upsilon[1];
        Scalar const uz = Scalar(0.5) * upsilon[2];

        // clang-format off
        J << o, o, o, h, o, o, o, o, o, o, h, o, o, o, o, o, o, h, o, o, o, o, o,
            o, i, o, o, o, uz, -uy, o, i, o, -uz, o, ux, o, o, i, uy, -ux, o;

        // clang-format on
        return J;
      }

      Scalar const c4 = sqrt(c3);
      Scalar const c5 = Scalar(1.0) / c4;
      Scalar const c6 = Scalar(0.5) * c4;
      Scalar const c7 = sin(c6);
      Scalar const c8 = c5 * c7;
      Scalar const c9 = pow(c3, -3.0L / 2.0L);
      Scalar const c10 = c7 * c9;
      Scalar const c11 = Scalar(1.0) / c3;
      Scalar const c12 = cos(c6);
      Scalar const c13 = Scalar(0.5) * c11 * c12;
      Scalar const c14 = c7 * c9 * omega[0];
      Scalar const c15 = Scalar(0.5) * c11 * c12 * omega[0];
      Scalar const c16 = -c14 * omega[1] + c15 * omega[1];
      Scalar const c17 = -c14 * omega[2] + c15 * omega[2];
      Scalar const c18 = omega[1] * omega[2];
      Scalar const c19 = -c10 * c18 + c13 * c18;
      Scalar const c20 = c5 * omega[0];
      Scalar const c21 = Scalar(0.5) * c7;
      Scalar const c22 = c5 * omega[1];
      Scalar const c23 = c5 * omega[2];
      Scalar const c24 = -c1;
      Scalar const c25 = -c2;
      Scalar const c26 = c24 + c25;
      Scalar const c27 = sin(c4);
      Scalar const c28 = -c27 + c4;
      Scalar const c29 = c28 * c9;
      Scalar const c30 = cos(c4);
      Scalar const c31 = -c30 + Scalar(1);
      Scalar const c32 = c11 * c31;
      Scalar const c33 = c32 * omega[2];
      Scalar const c34 = c29 * omega[0];
      Scalar const c35 = c34 * omega[1];
      Scalar const c36 = c32 * omega[1];
      Scalar const c37 = c34 * omega[2];
      Scalar const c38 = pow(c3, -5.0L / 2.0L);
      Scalar const c39 = Scalar(3) * c28 * c38 * omega[0];
      Scalar const c40 = c26 * c9;
      Scalar const c41 = -c20 * c30 + c20;
      Scalar const c42 = c27 * c9 * omega[0];
      Scalar const c43 = c42 * omega[1];
      Scalar const c44 = pow(c3, -2);
      Scalar const c45 = Scalar(2) * c31 * c44 * omega[0];
      Scalar const c46 = c45 * omega[1];
      Scalar const c47 = c29 * omega[2];
      Scalar const c48 = c43 - c46 + c47;
      Scalar const c49 = Scalar(3) * c0 * c28 * c38;
      Scalar const c50 = c9 * omega[0] * omega[2];
      Scalar const c51 = c41 * c50 - c49 * omega[2];
      Scalar const c52 = c9 * omega[0] * omega[1];
      Scalar const c53 = c41 * c52 - c49 * omega[1];
      Scalar const c54 = c42 * omega[2];
      Scalar const c55 = c45 * omega[2];
      Scalar const c56 = c29 * omega[1];
      Scalar const c57 = -c54 + c55 + c56;
      Scalar const c58 = Scalar(-2) * c56;
      Scalar const c59 = Scalar(3) * c28 * c38 * omega[1];
      Scalar const c60 = -c22 * c30 + c22;
      Scalar const c61 = -c18 * c39;
      Scalar const c62 = c32 + c61;
      Scalar const c63 = c27 * c9;
      Scalar const c64 = c1 * c63;
      Scalar const c65 = Scalar(2) * c31 * c44;
      Scalar const c66 = c1 * c65;
      Scalar const c67 = c50 * c60;
      Scalar const c68 = -c1 * c39 + c52 * c60;
      Scalar const c69 = c18 * c63;
      Scalar const c70 = c18 * c65;
      Scalar const c71 = c34 - c69 + c70;
      Scalar const c72 = Scalar(-2) * c47;
      Scalar const c73 = Scalar(3) * c28 * c38 * omega[2];
      Scalar const c74 = -c23 * c30 + c23;
      Scalar const c75 = -c32 + c61;
      Scalar const c76 = c2 * c63;
      Scalar const c77 = c2 * c65;
      Scalar const c78 = c52 * c74;
      Scalar const c79 = c34 + c69 - c70;
      Scalar const c80 = -c2 * c39 + c50 * c74;
      Scalar const c81 = -c0;
      Scalar const c82 = c25 + c81;
      Scalar const c83 = c32 * omega[0];
      Scalar const c84 = c18 * c29;
      Scalar const c85 = Scalar(-2) * c34;
      Scalar const c86 = c82 * c9;
      Scalar const c87 = c0 * c63;
      Scalar const c88 = c0 * c65;
      Scalar const c89 = c9 * omega[1] * omega[2];
      Scalar const c90 = c41 * c89;
      Scalar const c91 = c54 - c55 + c56;
      Scalar const c92 = -c1 * c73 + c60 * c89;
      Scalar const c93 = -c43 + c46 + c47;
      Scalar const c94 = -c2 * c59 + c74 * c89;
      Scalar const c95 = c24 + c81;
      Scalar const c96 = c9 * c95;

      J(0, 0) = o;
      J(0, 1) = o;
      J(0, 2) = o;
      J(0, 3) = -c0 * c10 + c0 * c13 + c8;
      J(0, 4) = c16;
      J(0, 5) = c17;
      J(1, 0) = o;
      J(1, 1) = o;
      J(1, 2) = o;
      J(1, 3) = c16;
      J(1, 4) = -c1 * c10 + c1 * c13 + c8;
      J(1, 5) = c19;
      J(2, 0) = o;
      J(2, 1) = o;
      J(2, 2) = o;
      J(2, 3) = c17;
      J(2, 4) = c19;
      J(2, 5) = -c10 * c2 + c13 * c2 + c8;
      J(3, 0) = o;
      J(3, 1) = o;
      J(3, 2) = o;
      J(3, 3) = -c20 * c21;
      J(3, 4) = -c21 * c22;
      J(3, 5) = -c21 * c23;
      J(4, 0) = c26 * c29 + Scalar(1);
      J(4, 1) = -c33 + c35;
      J(4, 2) = c36 + c37;
      J(4, 3) = upsilon[0] * (-c26 * c39 + c40 * c41) + upsilon[1] * (c53 + c57) +
                upsilon[2] * (c48 + c51);
      J(4, 4) = upsilon[0] * (-c26 * c59 + c40 * c60 + c58) +
                upsilon[1] * (c68 + c71) + upsilon[2] * (c62 + c64 - c66 + c67);
      J(4, 5) = upsilon[0] * (-c26 * c73 + c40 * c74 + c72) +
                upsilon[1] * (c75 - c76 + c77 + c78) + upsilon[2] * (c79 + c80);
      J(5, 0) = c33 + c35;
      J(5, 1) = c29 * c82 + Scalar(1);
      J(5, 2) = -c83 + c84;
      J(5, 3) = upsilon[0] * (c53 + c91) +
                upsilon[1] * (-c39 * c82 + c41 * c86 + c85) +
                upsilon[2] * (c75 - c87 + c88 + c90);
      J(5, 4) = upsilon[0] * (c68 + c79) + upsilon[1] * (-c59 * c82 + c60 * c86) +
                upsilon[2] * (c92 + c93);
      J(5, 5) = upsilon[0] * (c62 + c76 - c77 + c78) +
                upsilon[1] * (c72 - c73 * c82 + c74 * c86) +
                upsilon[2] * (c57 + c94);
      J(6, 0) = -c36 + c37;
      J(6, 1) = c83 + c84;
      J(6, 2) = c29 * c95 + Scalar(1);
      J(6, 3) = upsilon[0] * (c51 + c93) + upsilon[1] * (c62 + c87 - c88 + c90) +
                upsilon[2] * (-c39 * c95 + c41 * c96 + c85);
      J(6, 4) = upsilon[0] * (-c64 + c66 + c67 + c75) + upsilon[1] * (c48 + c92) +
                upsilon[2] * (c58 - c59 * c95 + c60 * c96);
      J(6, 5) = upsilon[0] * (c71 + c80) + upsilon[1] * (c91 + c94) +
                upsilon[2] * (-c73 * c95 + c74 * c96);

      return J;
    }

    /// Returns derivative of exp(x) wrt. x_i at x=0.
    ///
    SOPHUS_FUNC static Sophus::Matrix<Scalar, num_parameters, DoF>
    Dx_exp_x_at_0()
    {
      Sophus::Matrix<Scalar, num_parameters, DoF> J;
      Scalar const o(0);
      Scalar const h(0.5);
      Scalar const i(1);

      // clang-format off
      J << o, o, o, h, o, o, o,
           o, o, o, h, o, o, o,
           o, o, o, h, o, o, o,
           o, o, o, i, o, o, o,
           o, o, o, i, o, o, o,
           o, o, o, i, o, o, o;

      // clang-format on
      return J;
    }

    /// Returns derivative of exp(x).matrix() wrt. ``x_i at x=0``.
    ///
    SOPHUS_FUNC static Transformation Dxi_exp_x_matrix_at_0(int i)
    {
      return generator(i);
    }

    /// Group exponential
    ///
    /// This functions takes in an element of tangent space (= twist ``a``) and
    /// returns the corresponding element of the group SE(3).
    ///
    /// The first three components of ``a`` represent the translational part
    /// ``upsilon`` in the tangent space of SE(3), while the last three components
    /// of ``a`` represents the rotation vector ``omega``.
    /// To be more specific, this function computes ``expmat(hat(a))`` with
    /// ``expmat(.)`` being the matrix exponential and ``hat(.)`` the hat-operator
    /// of SE(3), see below.
    ///
    SOPHUS_FUNC static SE3<Scalar> exp(Tangent const &a)
    {
      using std::cos;
      using std::sin;
      Vector3<Scalar> const omega = a.template tail<3>();

      Scalar theta;
      SO3<Scalar> const so3 = SO3<Scalar>::expAndTheta(omega, &theta);
      Matrix3<Scalar> const Omega = SO3<Scalar>::hat(omega);
      Matrix3<Scalar> const Omega_sq = Omega * Omega;
      Matrix3<Scalar> V;

      if (theta < Constants<Scalar>::epsilon())
      {
        V = so3.matrix();
        /// Note: That is an accurate expansion!
      }
      else
      {
        Scalar theta_sq = theta * theta;
        V = (Matrix3<Scalar>::Identity() +
             (Scalar(1) - cos(theta)) / (theta_sq)*Omega +
             (theta - sin(theta)) / (theta_sq * theta) * Omega_sq);
      }

      return SE3<Scalar>(so3, V * a.template head<3>());
    }

    /// Returns closest SE3 given arbirary 4x4 matrix.
    ///
    template <class S = Scalar>
    SOPHUS_FUNC static enable_if_t<std::is_floating_point<S>::value, SE3>
    fitToSE3(Matrix4<Scalar> const &T)
    {
      return SE3(SO3<Scalar>::fitToSO3(T.template block<3, 3>(0, 0)),
                 T.template block<3, 1>(0, 3));
    }

    /// Returns the ith infinitesimal generators of SE(3).
    ///
    /// The infinitesimal generators of SE(3) are:
    ///
    /// ```
    ///         |  0  0  0  1 |
    ///   G_0 = |  0  0  0  0 |
    ///         |  0  0  0  0 |
    ///         |  0  0  0  0 |
    ///
    ///         |  0  0  0  0 |
    ///   G_1 = |  0  0  0  1 |
    ///         |  0  0  0  0 |
    ///         |  0  0  0  0 |
    ///
    ///         |  0  0  0  0 |
    ///   G_2 = |  0  0  0  0 |
    ///         |  0  0  0  1 |
    ///         |  0  0  0  0 |
    ///
    ///         |  0  0  0  0 |
    ///   G_3 = |  0  0 -1  0 |
    ///         |  0  1  0  0 |
    ///         |  0  0  0  0 |
    ///
    ///         |  0  0  1  0 |
    ///   G_4 = |  0  0  0  0 |
    ///         | -1  0  0  0 |
    ///         |  0  0  0  0 |
    ///
    ///         |  0 -1  0  0 |
    ///   G_5 = |  1  0  0  0 |
    ///         |  0  0  0  0 |
    ///         |  0  0  0  0 |
    /// ```
    ///
    /// Precondition: ``i`` must be in [0, 5].
    ///
    SOPHUS_FUNC static Transformation generator(int i)
    {
      SOPHUS_ENSURE(i >= 0 && i <= 5, "i should be in range [0,5].");
      Tangent e;
      e.setZero();
      e[i] = Scalar(1);

      return hat(e);
    }

    /// hat-operator
    ///
    /// It takes in the 6-vector representation (= twist) and returns the
    /// corresponding matrix representation of Lie algebra element.
    ///
    /// Formally, the hat()-operator of SE(3) is defined as
    ///
    ///   ``hat(.): R^6 -> R^{4x4},  hat(a) = sum_i a_i * G_i``  (for i=0,...,5)
    ///
    /// with ``G_i`` being the ith infinitesimal generator of SE(3).
    ///
    /// The corresponding inverse is the vee()-operator, see below.
    ///
    SOPHUS_FUNC static Transformation hat(Tangent const &a)
    {
      Transformation Omega;
      Omega.setZero();
      Omega.template topLeftCorner<3, 3>() =
          SO3<Scalar>::hat(a.template tail<3>());
      Omega.col(3).template head<3>() = a.template head<3>();

      return Omega;
    }

    /// Lie bracket
    ///
    /// It computes the Lie bracket of SE(3). To be more specific, it computes
    ///
    ///   ``[omega_1, omega_2]_se3 := vee([hat(omega_1), hat(omega_2)])``
    ///
    /// with ``[A,B] := AB-BA`` being the matrix commutator, ``hat(.)`` the
    /// hat()-operator and ``vee(.)`` the vee()-operator of SE(3).
    ///
    SOPHUS_FUNC static Tangent lieBracket(Tangent const &a, Tangent const &b)
    {
      Vector3<Scalar> const upsilon1 = a.template head<3>();
      Vector3<Scalar> const upsilon2 = b.template head<3>();
      Vector3<Scalar> const omega1 = a.template tail<3>();
      Vector3<Scalar> const omega2 = b.template tail<3>();

      Tangent res;
      res.template head<3>() = omega1.cross(upsilon2) + upsilon1.cross(omega2);
      res.template tail<3>() = omega1.cross(omega2);

      return res;
    }

    /// Construct x-axis rotation.
    ///
    static SOPHUS_FUNC SE3 rotX(Scalar const &x)
    {
      return SE3(SO3<Scalar>::rotX(x), Sophus::Vector3<Scalar>::Zero());
    }

    /// Construct y-axis rotation.
    ///
    static SOPHUS_FUNC SE3 rotY(Scalar const &y)
    {
      return SE3(SO3<Scalar>::rotY(y), Sophus::Vector3<Scalar>::Zero());
    }

    /// Construct z-axis rotation.
    ///
    static SOPHUS_FUNC SE3 rotZ(Scalar const &z)
    {
      return SE3(SO3<Scalar>::rotZ(z), Sophus::Vector3<Scalar>::Zero());
    }

    /// Draw uniform sample from SE(3) manifold.
    ///
    /// Translations are drawn component-wise from the range [-1, 1].
    ///
    template <class UniformRandomBitGenerator>
    static SE3 sampleUniform(UniformRandomBitGenerator &generator)
    {
      std::uniform_real_distribution<Scalar> uniform(Scalar(-1), Scalar(1));
      return SE3(SO3<Scalar>::sampleUniform(generator),
                 Vector3<Scalar>(uniform(generator), uniform(generator),
                                 uniform(generator)));
    }

    /// Construct a translation only SE3 instance.
    ///
    template <class T0, class T1, class T2>
    static SOPHUS_FUNC SE3 trans(T0 const &x, T1 const &y, T2 const &z)
    {
      return SE3(SO3<Scalar>(), Vector3<Scalar>(x, y, z));
    }

    static SOPHUS_FUNC SE3 trans(Vector3<Scalar> const &xyz)
    {
      return SE3(SO3<Scalar>(), xyz);
    }

    /// Construct x-axis translation.
    ///
    static SOPHUS_FUNC SE3 transX(Scalar const &x)
    {
      return SE3::trans(x, Scalar(0), Scalar(0));
    }

    /// Construct y-axis translation.
    ///
    static SOPHUS_FUNC SE3 transY(Scalar const &y)
    {
      return SE3::trans(Scalar(0), y, Scalar(0));
    }

    /// Construct z-axis translation.
    ///
    static SOPHUS_FUNC SE3 transZ(Scalar const &z)
    {
      return SE3::trans(Scalar(0), Scalar(0), z);
    }

    /// vee-operator
    ///
    /// It takes 4x4-matrix representation ``Omega`` and maps it to the
    /// corresponding 6-vector representation of Lie algebra.
    ///
    /// This is the inverse of the hat()-operator, see above.
    ///
    /// Precondition: ``Omega`` must have the following structure:
    ///
    ///                |  0 -f  e  a |
    ///                |  f  0 -d  b |
    ///                | -e  d  0  c
    ///                |  0  0  0  0 | .
    ///
    SOPHUS_FUNC static Tangent vee(Transformation const &Omega)
    {
      Tangent upsilon_omega;
      upsilon_omega.template head<3>() = Omega.col(3).template head<3>();
      upsilon_omega.template tail<3>() =
          SO3<Scalar>::vee(Omega.template topLeftCorner<3, 3>());

      return upsilon_omega;
    }

  protected:
    SO3Member so3_;
    TranslationMember translation_;
  };

  template <class Scalar, int Options>
  SE3<Scalar, Options>::SE3() : translation_(TranslationMember::Zero())
  {
    static_assert(std::is_standard_layout<SE3>::value,
                  "Assume standard layout for the use of offsetof check below.");
    static_assert(
        offsetof(SE3, so3_) + sizeof(Scalar) * SO3<Scalar>::num_parameters ==
            offsetof(SE3, translation_),
        "This class assumes packed storage and hence will only work "
        "correctly depending on the compiler (options) - in "
        "particular when using [this->data(), this-data() + "
        "num_parameters] to access the raw data in a contiguous fashion.");
  }
} // namespace Sophus

namespace Eigen
{

  /// Specialization of Eigen::Map for ``SE3``; derived from SE3Base.
  ///
  /// Allows us to wrap SE3 objects around POD array.
  template <class Scalar_, int Options>
  class Map<Sophus::SE3<Scalar_>, Options>
      : public Sophus::SE3Base<Map<Sophus::SE3<Scalar_>, Options>>
  {
  public:
    using Base = Sophus::SE3Base<Map<Sophus::SE3<Scalar_>, Options>>;
    using Scalar = Scalar_;
    using Transformation = typename Base::Transformation;
    using Point = typename Base::Point;
    using HomogeneousPoint = typename Base::HomogeneousPoint;
    using Tangent = typename Base::Tangent;
    using Adjoint = typename Base::Adjoint;

    // LCOV_EXCL_START
    SOPHUS_INHERIT_ASSIGNMENT_OPERATORS(Map);
    // LCOV_EXCL_STOP

    using Base::operator*=;
    using Base::operator*;

    SOPHUS_FUNC Map(Scalar *coeffs)
        : so3_(coeffs),
          translation_(coeffs + Sophus::SO3<Scalar>::num_parameters) {}

    /// Mutator of SO3
    ///
    SOPHUS_FUNC Map<Sophus::SO3<Scalar>, Options> &so3() { return so3_; }

    /// Accessor of SO3
    ///
    SOPHUS_FUNC Map<Sophus::SO3<Scalar>, Options> const &so3() const
    {
      return so3_;
    }

    /// Mutator of translation vector
    ///
    SOPHUS_FUNC Map<Sophus::Vector3<Scalar, Options>> &translation()
    {
      return translation_;
    }

    /// Accessor of translation vector
    ///
    SOPHUS_FUNC Map<Sophus::Vector3<Scalar, Options>> const &translation() const
    {
      return translation_;
    }

  protected:
    Map<Sophus::SO3<Scalar>, Options> so3_;
    Map<Sophus::Vector3<Scalar>, Options> translation_;
  };

  /// Specialization of Eigen::Map for ``SE3 const``; derived from SE3Base.
  ///
  /// Allows us to wrap SE3 objects around POD array.
  template <class Scalar_, int Options>
  class Map<Sophus::SE3<Scalar_> const, Options>
      : public Sophus::SE3Base<Map<Sophus::SE3<Scalar_> const, Options>>
  {
  public:
    using Base = Sophus::SE3Base<Map<Sophus::SE3<Scalar_> const, Options>>;
    using Scalar = Scalar_;
    using Transformation = typename Base::Transformation;
    using Point = typename Base::Point;
    using HomogeneousPoint = typename Base::HomogeneousPoint;
    using Tangent = typename Base::Tangent;
    using Adjoint = typename Base::Adjoint;

    using Base::operator*=;
    using Base::operator*;

    SOPHUS_FUNC Map(Scalar const *coeffs)
        : so3_(coeffs),
          translation_(coeffs + Sophus::SO3<Scalar>::num_parameters) {}

    /// Accessor of SO3
    ///
    SOPHUS_FUNC Map<Sophus::SO3<Scalar> const, Options> const &so3() const
    {
      return so3_;
    }

    /// Accessor of translation vector
    ///
    SOPHUS_FUNC Map<Sophus::Vector3<Scalar> const, Options> const &translation()
        const
    {
      return translation_;
    }

  protected:
    Map<Sophus::SO3<Scalar> const, Options> const so3_;
    Map<Sophus::Vector3<Scalar> const, Options> const translation_;
  };
} // namespace Eigen

#endif
